An Introduction to Quantum and Vassiliev Knot Invariants by David M. Jackson & Iain Moffatt

Author:David M. Jackson & Iain Moffatt
Language: eng
Format: epub
ISBN: 9783030052133
Publisher: Springer International Publishing

(11.4)

For the second part, given and hence there is a unique that is, induced by through (11.4). Thus, we can define by

Now, consider Then, and Also, Let and . Then,

for all Thus, Similarly, for all Thus, is linear.

Finally, the element-wise action follows from the definition of and its uniqueness.

As an example of calculating the value of , let be the coefficient of in the Alexander–Conway polynomial, C(z) (Definition 2.​30). We already know that is a Vassiliev invariant by Theorem 10.​15. (Note that in this example C is the Alexander–Conway polynomial, rather than a chord diagram as it is in the rest of this chapter.) We determine explicitly by using any preimage of under We have . Then, by repeated application of the Vassiliev relation, the Vassiliev resolution of the singular knot is

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